Diblock copolymer lamellae on sinusoidal and fractal surfaces

A scaling analysis of equilibrium orientation of diblock copolymer molecules on fractal surfaces and a brief comparison with a particular experiment is presented in this paper. This work is motivated by a recent experimental finding that a diblock copolymer film of polystyrene-PMMA, when deposited on a rough substrate, can orient its lamellae from a parallel to a perpendicular configuration depending on the topographical characteristics of the substrate surface. It was found that the RMS height itself is not enough to effect the equilibrium configuration, but the fractal dimension of the surface is also important. In general, the orientation of lamellae is a function of the the power spectral density (PSD) curves of the underlying substrate surface. Assuming the diblock lamellae to behave like an Alexander-deGennes brush, we obtain the free energy expressions for this brush in both parallel and perpendicular orientations in various asymptotic regimes. Comparison of their free energy expressions predicts the equilibrium configuration. By examining the PSD curves and using our scaling results, we are able to qualitatively explain some aspects of the experimental observations regarding the equilibrium orientation of the diblock copolymer lamellae on rough surfaces.